# A Main Lemma for Continuous Solutions

# A Main Lemma for Continuous Solutions

This chapter introduces the Main Lemma that implies the existence of continuous solutions. According to this lemma, there exist constants *K* and *C* such that the following holds: Let ϵ > 0, and suppose that (*v*, *p*, *R*) are uniformly continuous solutions to the Euler-Reynolds equations on ℝ x ³, with *v* uniformly bounded⁷ and *suppR* ⊆ *I* x ³ for some time interval. The Main Lemma implies the following theorem: There exist continuous solutions (*v*, *p*) to the Euler equations that are nontrivial and have compact support in time. To establish this theorem, one repeatedly applies the Main Lemma to produce a sequence of solutions to the Euler-Reynolds equations. To make sure the solutions constructed in this way are nontrivial and compactly supported, the lemma is applied with *e*(*t*) chosen to be any sequence of non-negative functions whose supports are all contained in some finite time interval.

*Keywords:*
continuous solution, Euler-Reynolds equations, theorem, non-negative function, finite time interval

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